23. Fresnel Equations

• EM Waves at boundaries

• Fresnel Equations: Reflection and Transmission Coefficients

• Brewster’s Angle

• Total Internal Reflection (TIR)• Evanescent Waves

• The Complex Refractive Index• Reflection from Metals

2 2

2 2

2 2 2

2 2 2

cos sin

cos sin

cos sin

c

:

os sin

rTE

rTM

E nrE n

E n nr

r reflection coef

E n n

ficient

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− + −= =

+ −

2 2

2 2 2

2cos

cos sin2 cos

cos s

:

in

tTE

tTM

t transmission coefficieE

tE nE ntE

n

n

t

n

θ

θ θθ

θ θ

= =+ −

= =+ −

n2

θE

Et

Erθr

θt

n1

We will derive the Fresnel equations

1

2

nn

nnn

incident

dtransmitte =≡

EM Waves at an Interface

( )( )( )

: exp

: exp

: exp

oi

or

i i i

r r

tt

r

tot

k

k

k

Incident beam E i r t

Reflected beam E i r t

Tran

E

E

smitted beam E i tE r

ω

ω

ω

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦

rr r r

r r

r rr

r

r

r1 0

1 0

2 0

i

r

t

k n k

k n k

k n k

=

=

=

r

r

r

r

TE mode

n1

n2

iθ

n1

n2

TM mode

Note the definition of the positive E-field directions in both cases.

n2

n1

iEr

rEr

tEr

ikr

rkr

tkr

oiEr

orEr

otEr

EM Waves at an Interface

( ), ,

points

, the tangential component must be equal on both sides of the inter

At the boundary between the two media the x y plane all waves must exist simultaneouslyand .Therefore for all time t and fo

fall

acboundary r on th te

er e in

−

r ,rface

( )( )( )

exp

exp

exp

:

:

:

i oi i

r or

i

r

t ot t

r

t

E E i r t

E E i r

Incident beam

Reflected beam

Transmitted bea

k

k

km

t

E E i r t

ω

ω

ω

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦

r r r

r r r

r r r

r

r

r

( ) ( ) ( ) : Phase matching at the boundary

,

!

:

i i r r t t

the only way that this can be true over the entire interfac

k r t k r t k r t

e and for all t is iAssuming that the wave amplitudes are constan

ft

ω ω ω⇒ ⋅ − = ⋅ − = ⋅ −r r rr r r

( ) ( ) ( )exp exp expoi i i or r r ot t

i r t

tn E i k r t n E i k r t n E i k

n E n E n

r t

E

ω ω ω⎡ ⎤ ⎡ ⎤ ⎡ ⎤× ⋅ − + ×

× +

⋅ − = × ⋅ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

× = ×r r rr r r

r r r) )

r r r) ) )

)n2

n1

iEr

rEr

tEr

ikr

rkr

tkr

oiEr

orEr

otEr

n̂

rr

EM Waves at an Interface

Normal

xikr

rkr

tkr

1 0n k

2 0n k

rr

1 0

1 0

2 0

i

r

t

k n k

k n k

k n k

=

=

=

r

r

r

( ) ( ) ( )Phase matching condition:

i i r r t tk r t k r t k r tω ω ω⋅ − = ⋅ − = ⋅ −r r rr r r

(Frequency does not change at the boundary!)

0,

t

i t

i r

r

At r this results int t tω ω ω

ω ω ω

== =

⇒ = =

r

, ,

, ,

,

and .

, .i r

i r t

i r

t

t

k r constant

the equation for a plane perpendicular to k

k k and k are coplanar in the plane of incidence

r

⇒ ⋅

⇒

=

→

r r

r

r

r

r r

(Phases on the boundary does not change

,

!)

0

i r t

At t this

k r

results i

k k

n

r r⋅ =

=

=⇒ ⋅ ⋅r r rr r r

n2

n1

iEr

rEr

tEr

ikr

rkr

tkr

rr

EM Waves at an Interface

co

0,

,

sin si

ns

n

,

sin

ta

s

n

n

t

i

i r i i r r

i

ii r

t

r i ri

r

At t

Considering the relation for the incident and reflected beams

k r k r k r k r

Since the incident and reflected beams are in t

k r k r

he same mediumnk k

k r

c

θ θ

ω θ θ θ θ

=

⋅ = ⋅ ⇒ =

= = ⇒ = ⇒

⋅ = ⋅ =

=

⋅ =

r rr r

r r rr r r

: law of reflection

,

sin sin

sin sin : law of refractio

,

n

i t i

i i t

i t t

i tt ti

Considering the relation for the incident and transmitted beams

k r k r k r k r

But the incident and transmitted beams are in different median nk k

c cn n

θ θ

ω θω θ=

⋅ = ⋅ ⇒ =

= = ⇒

r rr r

iθ

rθ

tθ

Normal

xikr

rkr

tkr

1 0n k

2 0n k

rr

n2

θι θr

θt

n1 x

iEr

rEr

tEr

ikr

rkr

tkr

Development of the Fresnel Equations

cos co

' ,

s co

:

si r t

i i r r t t

E E EB B B

From Maxwell s EM field theorywe have the boundary conditions at the interface

Th tangential

components of both E and B are equal onboth sides o

e above co

f the i

nditions imply that th

for the T

e

E case

θ θ θ+ =

− =

r r

0

cos cos

.,

c

:

os

.

i i r r t t

i t

i r t

We have alsoassumed that as is true formost dielectric materia

nterface

E E EB

For the TM mod

B B

e

ls

θ θ θ

μ μ μ

+ =− + = −

≅ ≅

TE-case

TM-case

Development of the Fresnel Equations

1

1 1

1

2

2

1

:

cos cos

:

cos cos c

v

s

c

o

os

i i r

i r t

i

r t t

i

i r r t t

cRecall that E B Bn

Let n refractive index of incident mediumn refractive index of refracting me

For the TM m

diu

For the TE mod

nEBc

ode

E E

e

E E En E n E E

n

m

n

En E E

θ θ

θ θ θ

θ

=⎛ ⎞= = ⇒⎜ ⎟⎝ ⎠

==

−

+

+

=

+

−

=

=

2r tn E= −

TE-case

TM-case

n1

n2

n1

n2

Development of the Fresnel Equations

2

1

sin sin

cos cos:cos cos

cos cos:cos co

:

s

cos

i trT

t

i t

Ei i t

i trTM

i i t

from each set of e

nETE case rE n

nETM ca

quationsand solving for the refleEliminating

ction coefficient we obtain

wher

E

nne

We know t

se rE

n

a

n

t

n

n

h

θ

θ θ

θθ

θ θθ θ

θ

θ−

= =+

− += =

=

=

+

22 2 2

2

sin1 sin 1 sinit t in n n

nθθ θ= − = − = −

TE-case

TM-case

n1

n2

n1

n2

TE-case

TM-case

n1

n2

n1

n2

Now we have derived the Fresnel Equations

2 2

2 2

2 2 2

2 2 2

cos sin

cos si

:

:

:

:

n

cos sin

cos sin

i irTE

i i i

i irTM

i i i

Substituting we obtain t reflection coefficiehe Fresnel equa nts r

nErE n

n nErE n

transmission c

tions for

TE case

TM case

For the

T

oeffici n t

n

e t

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− + −= =

+ −

2 2

2 2 2

2coscos sin

2 cos

: 1:

cos sin

1

:

:

TE TE

TM T

t iTE

i i i

t iTM

i

M

i i

E case

TM ca

EtE

TE t rTM n

nE ntE n

t

nse

r

θ

θ θθ

θ θ

= =+ −

= =

+= −

+ −

=

1

2

nnn ≡

These just mean the boundary conditions.

::i r t

i r t

For the TE case E E EFor the TM mode B B B

+ =− + = −

Power : Reflectance (R) and Transmittance (T).

1

,,

:

:

tr

i i

i r t

R and T are the ratios of reflected and transmitThe quantitiesThe ratiosrespectively to

ted powers

PPR TP P

R T

r and t are ratios of electric field amplitudes

From conservation of ener

the incident power

P P P

We can

gy

= =

= += + ⇒

21 0

20 00

:

cos cos

cos cos cos

1

cos

1 c22

i i i r r r t t t

i i r r

i i r r t

i i r r t

i

t

t

t t

iexpress the power in each of the fie

n terms of the product of an irradiance and areaP I A P I A P I A

I

lds

I A I A I A

But n c

I I

I n c

I A I I A

E

A

E

θ θ θ

ε

θ θ θ

ε

=⇒=+

=+

⇒

= = =

+

= 2 21 0 0 2 0 0

2 2 2 20 2 0 0

2 220 0

2 20 0

02 2 2 20 1 0 0 0

1 1os cos cos2 2

cos cos 1cos

cos coscos

co

s

s

co

i

r t t t

i i

r r t t

r t t r t t

i i i i

i

i i

i

n cE n cE

E n E E E

E ER r T n

n R TE

E

n E E

nE

E

θ ε θ ε θ

θ θθ

θ θθ θ

θ⎛ ⎞ ⎛ ⎞

=

= +

⎛ ⎞⇒ = + = + = +⎜ ⎟

⎝ ⎠

= = =⎜ ⎟ ⎜⎝ ⎝

⇒⎠ ⎠

2t⎟2

2

coscos*

coscos

*

tnttnT

rrrR

i

t

i

t⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

==

θθ

θθ

2

2

coscos_cos cos

out out outout out

in in in in in

n EIPower ratioI n E

θθθ θ

⎛ ⎞= = ⎜ ⎟

⎜ ⎟⎝ ⎠

A

23-2. External and Internal Reflection

( )2 1

2 22 1

,

/ 1 sin 0

are always realTE TM

n n n n

r

n n

θ

⇒

⇒

>

= > ⇒ − ≥

⇒

rTE

t

n=1.50

Brewster’s angle (or, polarizing angle)(No reflection of TM mode)( ) 1

:

tan 0TM p p

fNo or tht e TM case

n

e

r whenθ θ θ −=⇒ = =

External Reflection

, 0TE TMr >

, 0TE TMr <

, 0TE TMt >2 2

2

2 2

2

2

2 2 2

cos sin

c

cos sin

co os sins sini ii

TM

i

iTE

i ii

n nr

n

nr

nn

θ θ

θ

θ θ

θ θθ

− + −=

+ −

− −=

+ −

, If 0 then there are no phase changes after reflection.TE TMr⇒ >

,

, , ,

If 0 then there are always ( 180 ) phase changes.

TE TM

TE TM TE TE Mi

TM T

r

er r rπ

π⇒ < =

→ = =−

o

[ ]2 1/ 1n n n= >

rTM

( )2 2,

,

, ,

, If sin 0, =1,

=1

(- ~ ) phase change may occur after reflect

ion

!TE TTE TM

TE TM

i iTE TM TE

M

TM

BUT r are compn r

r

r r e e

lex

φ φ

θ

φ π π

⇒ − <

→

→ = =

→ +

( )1tan

: 0p

TM

No Brewster's angle n

for the TM

e

case

t

r

θ −=

=

critical angle

Internal Reflection

2 2

2

2 2

2

2

2 2 2

cos sin

c

cos sin

co os sins sini ii

TM

i

iTE

i ii

n nr

n

nr

nn

θ θ

θ

θ θ

θ θθ

− + −=

+ −

− −=

+ −

, 0TE TMr >

, 0TE TMr <

Total internal reflection (TIR) when θ > θc

TIR region

( ) ( )21

2 2

2 1

2 2

/ 1

sin 0, , sin 0

n n n

n o

n n

r nθ θ

⇒ = <

⇒ − > − <

>

( )2 2,

2 1

If sin 0, =1

sin ( / )TE TM

c

n r

n n n

θ

θ

⇒ − =

→ = =

( )2 2,

,

,

If sin 0, are always real

If 0 then there are no phase changes after reflection.

If 0 then there are ( 180 ) phase changes.

TE TM

TE TM

TE TM

n r

r

r

θ

π

⇒ − >

→ >

→ < = o

[ ]2 1/ 1n n n= <

Derivation of Brewster’s Angle

( )

4 2 2 2

2 2 2

4 2 2 2

1

2 2 2

2 2 2 2

c

( ) :

cos sin

cos s

os sin0

cos sin

1.50, 56.

in

( 1) os si

tan

n 0

31

p

p p

p pTM p

p p

p

p

p p

p p

Brewster's angle for polarizing ang

n nr

n n

For n

le

n n

n n

n n c

n

θ

θ θ

θ θ

θ θθ

θ

θ

θ

θ

θ

θ

−

⇒ = −

− +

− + −= =

⎡ ⎤= − − =⎣

+

= =

=

⎦

−

⇒

°

externalreflectioninternal

reflection

θp θp

θc

R

External & Internal reflections, but TM-polarization only

1 : sin n nc <=θ

or nn np 11 : tan <>=θ

TE & TM polarizations, but Internal reflection only

Brewster ‘s angle :

Critical angle :

TE TM

Total Internal Reflection (TIR) 2

1

1 , ,and for both (TE and TM) cases.

sin t

: 1

1otal internal r

* 1

eflection( )

cos

rTE

i

c

nInternal reflection nn

r R rr is a complex

call

numb

eFor n TId

er

ErE

r

Rθ θ −

⇒

= <

= = =

≥

=

=

⇒

=2 2

2 2

2 2 2

2 2 2

sin

cos sin

cos sin

cos sin

i i

i i

i irTM

i i i

i n

n

n nErE n

i

i n

i

θ θ

θ θ

θ θ

θ θ

− −

+ −

− + −= =

+ −

internalreflection

R

r

θc

Complex value

R = 1

rTEn=1.50

, 0TE TMr >

, 0TE TMr <

, 0TE TMt >

23-3. Phase changes on reflection

,

,

,

18

,

,0

) 0

0

.0 (

TE T

TE TM

TE TM

Mthe phase

r is always a real numb

the

er for ex

phase sh

shift is fo

ift is

ternal reflection

then

an for r

r

d

r

π

° >

° = <

TE TM

Phase shift after External Reflection

For TE case, π phase shift for all incident angles For TM case, π phase shift for θ < θpNo phase shift for θ > θp

External Reflection External Reflection

External reflection

rTM

cθθ >:

Complex value

In TIR region

Phase shift after Internal Reflection Internal reflection

1 0 for sin is complex in TIR region where

TE TE

TE c

TE ci i

TE TE

r nr

r r e eφ φ

θ θθ θ

−⇒ > < =⇒ >

→ = =

For TM case, no phase shift for θ < θpπ phase shift for θp < θ < θc

φTM(θ) phase shift for θ > θc

For TE case, no phase shift for θ < θc

φTE(θ) phase shift for θ > θc

TIR TIR

1 0 for tan

0 for

TM

TM p

TM p c

iTM TM TM TM

r n

r

r r e rφ

θ θ

θ θ θ

φ π

−⇒ > < =

⇒ < < <

→ = − = → =

is complex in TIR region where

TM TM

TM ci i

TM TM

r

r r e eφ φ

θ θ⇒ >

→ = =

2

:

cos sin sintancos sin

2

( )

s

co

ii i

i

c TIWhen then and for both the TE and TM cases has the form

a ib i e br e ea ib i

R case r is complex

e a

αα φ

α

α α α φ ααα

θ θ

α α

−−

+

− −= = = =

≥

== = −= ⇒+ +

2 2

2 2

2 2

2

1

2 2

2

cos sin:

cos sin

cos sin

sintan t

( ).

sin2 tan

co

an2 cos

s

i irTE

i i i

i i

iT

i

E

TEi

is the phase shift on total internal reflection TIR

n

i nETE case rE i n

a b n

n

A simi

φ

θ θ

θ θ

θ θ

θφ

θ

θφαθ

−

− −= =

+ −

= = −

−⎛ ⎞

⎛ ⎞−⎜ ⎟= −⎜ ⎟

⇒ = − =⎜⎝ ⎠

⎝ ⎠

⎟

2 21

2

sin2 tan

cos

:

iTM

i

lar analysis for the TM case giv

n

es

nθφ π

θ−⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

Internal reflection

: i cθ θ>

: i cθ θ>

TIR(Complex r )

Phase shifts on total Internal Reflection for both TE- and TM-cases

2 21

2

2 21

sin2 tan

cos

sin2 tan

cos

iTM

iTE

nn

n

θφ π

θ

θφ

θ

−

−

⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

Internal reflection

Complex value

Therefore, after TIR is ……….. ,TE TMr

,TE TMr

,TE TMφ

2 2

2 2

2 2 2

2 2 2

cos sin

cos sin

cos sin

cos sin

i irTE

i i i

i irTM

i i i

nErE n

n n

i

i

in n

i

ErE

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− + −= =

+ −

( )cincident θθ > case TIRFor

Internal reflectionTIR(Complex r )

Summary of Phase Shifts on Internal Reflection

'p

'p c

2 21

c2

0 <

( 180 ) <

sin2 tan <

cos

TM

i nn

θ θ

φ π θ θ θ

θπ θ θ

θ−

⎧⎪⎪⎪⎪= = <⎨⎪ ⎛ ⎞−⎪ ⎜ ⎟−⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

o

o

p

p c

c

0 <

<

0 TM TE

θ θ

φ φ φ π θ θ θ

θ θ

⎧=⎪

Δ = − = <⎨⎪> <⎩

o

o

c

2 21

c

0 <

sin2 tan >

cosTE i n

θ θ

φ θθ θ

θ−

⎧⎪⎪ ⎛ ⎞= −⎨ ⎜ ⎟−⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

o

φΔTMφ

TEφ

Fresnel Rhomb

Linearly polarized light (45o)

CircularlyPolarized

light

3 53 1.54

After two consequentive TIRs, 3 2

2

TM TE i

TM TE

TM TE

Note near when n

Quarter wave retarder

πφ φ θ

πφ φ

πφ φ φ

− = = =

→

→ − =

→ Δ = − =

→ −

o

φΔTMφ

TEφ

Linearly polarized light

(45o)

CircularlyPolarized

light

69 ???2

2

TM TE i

TM TE

Note near when n

Quarter wave retarder

πφ φ θ

πφ φ φ

− = = =

→ Δ = − =

→ −

oφΔTMφ

TEφ

Quarter-wave retardation after TIR

n

23-5. Evanescent Waves at an Interface

( )( )( )

( )

: exp

: exp

: exp

exp

si

:

n

i oi i i

r or r r

t ot t t

t ot t t

t t t

Incident beam E E i k r t

Reflected beam E E i k r t

Transmitted beam E E

For the transmitted

i k

bea

r t

E E i k r

k

m

t

r k

ω

ω

ω

ω

θ

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

⋅ =

rr r r

rr r r

rr r r

r r

r r )( ) ( )( )

2

22

cos

sin cos

sin, cos 1 sin 1

, :

sin

sinc

( )

o s 1it

t t

t t t

it t

iWhen n

x k z x x zz

k x z

Butn

th

i a purely imaginary numbe

total internal reflection

r

en

n

θ

θ

θθ

θ

θθ

θ

θ

+ ⋅ +

= +

=

= −

− −

⇒

=

>

)) )

Evanescent Waves at an Interface( )

2

2 2

sin

sin 1

,:

sin

:

sin sin21 1

:

tt t

it

i

i

t

t

iwith an TIR condition n

i z

For the transmitted beamwe can write the phase factor as

k r k xn

Defining the coefficient

kn n

We can write the transmitted wa s

n

v

E

e a

θ

α

θ θπαλ

θ

θ⎛ ⎞⋅ = +⎜ ⎟

⎜ ⎟⎝ ⎠

=

−

− −

=

=

>

r r

( )0sin ep

.

xx pe t tt z

amplitude will decay rapidThe evanescent waveas it penetrates into the lower refractive ind

ly

k xE i t

ex med

n

ium

θ ω α⎡ ⎤⎛ ⎞− −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Note that the incident and reflection wavesform a standing wave in x direction

n2

n1

n1 > n2 h

( )0sinexp expt t

t tk xE E i t z

nθ ω α⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

x

z

2

2

1 1sin2 1

t ot

i

E Ee

h

n

λα θπ

⎛ ⎞= ⇒ = =⎟⎝ ⎠

−⎜Penetration depth:

Frustrated TIR

Tp = fraction of intensity transmitted across gap

Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).

d

n1=n2=1.5171.65

d/λ

Frustrated Total Internal Reflectance

Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).

Pellin-Broca prism

d = 1 ~ λ: changing the reflectanceRotation: changing the wavelength resonant at θB

d

23-6. Complex Refractive Index

2 2 2

0

0

( ) :

1

1

2R I R I

R IFor a material with conductivity

n i n n i n

n i n i n

n

σε ω

σ

σε ω

⎛

⎛ ⎞= + = −

⎞= + = +⎜ ⎟

⎝ ⎠

+⎜ ⎟⎝ ⎠

%

%

2

0

2

02 2

2 4 2

0 0

2

0

2

022

:

1 2

1 02 2

:

1 1

2

1 1 42

42

2 2

RI

I

R I R I

I I II

I

Solving for the real and imaginary components we obtain

n n n n

n n nn

From the quadratic solution we obt

nn

n

ain

n

σε ω

σ σε ω ε ω

σε

σε

σω

ω

ε ω

− = = ⇒

⎛ ⎞ ⎛ ⎞⇒ − = ⇒ − − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞± +

=

⎛ ⎞+ + ⎜ ⎟

⎝⎜⎝ =⇒ ⎠

⎟⎠=

.IWe need to take the positive root because n is a real number

Complex Refractive Index

( )( ) ( )

( )

0

0

0

exp

ˆexp

eˆ xexp pRk

R k

I

I

Substituting our expression for the complex refractive index back intoour expression for the electric field we obtain

E E i k r t

E i n i n u r tc

E ni u r tc

nc

ω

ω

ω

ω

ω

⎡ ⎤= ⋅ −⎣

⎧ ⎫⎡ ⎤⋅ − −

⎦⎧ ⎫⎡ ⎤= + ⋅ −⎨ ⎬⎢ ⎥

⎨ ⎬⎢ ⎥⎣ ⎦

⎣ ⎦⎭

⎩

⎩

=⎭

rr

r

r r

r r

r( )

..

ˆ

/

k

R

The first exponential term is oscillatoryThe EM wave propagates w

u r

The second exponential has a r

ith a ve

eal arg

locity of

ument (absor .

n c

bed)

⎡ ⎤⋅⎢ ⎥⎣ ⎦

r

Complex Refractive Index

( )* *0 0

0

..

ˆ2exp

ˆ2exp

I k

I k

The second term leads to absorption of the beam in metals due to inducinga current in the medium This causes the irradiance to decrease as the wavepropagates through the medium

n u rI EE E E

c

n uI I

ω

ω

⋅⎡ ⎤≡ = −⎢ ⎥

⎣ ⎦

= −

rr r r r

( ) ( )0 ˆexp

2 4:

k

I I

rI u r

c

The absorption coefficient is defined n ncω πα

α

λ

⋅⎡ ⎤= − ⋅⎡

= =

⎤⎢ ⎥ ⎣ ⎦⎣ ⎦

rr

( ) ( )0 ˆex ˆexpp Ik k

R n uni u rc

rtc

E E ω ω⎧ ⎫⎡ ⎤⋅ −⎨ ⎬⎢ ⎥⎡ ⎤− ⋅=

⎣ ⎦⎩ ⎢ ⎥⎣ ⎦⎭

rr r r

23-7. Reflection from Metals

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos sin

:

i irTE

i i i

i irTM

i i i

Reflection from metals is analyzed

ETE case rE

nETM cas

by substituting the complex refractive index n i

e rE n

S

n the Fresnel

n

n

equa

ub

ti

n

n

n

o s

s

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− + −= =

+ −

%

%

%%

%

% %

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2

2 2 2

2 2 2 2 2

2 2 2 2 2

:

cos sin 2:

cos sin 2

2 cos sin 2:

2 cos sin 2

i R I i R Ir

i i R I i R I

R I R I i R I i R Ir

i R I R I i R

R I

I i R I

TE case

TM case

tituting we obtain

n n i n nErE n n i n n

n n i n n n n i n nErE n n i n n n n

n n i

i n n

n

θ θ

θ θ

θ θ

θ θ

− − − += =

+ − − +

⎡ ⎤− − + + − − +⎣ ⎦= =⎡ ⎤− + + − +

=

⎣

+

−⎦

%

Reflectance

θi

Reflection from Metals at normal incidence (θi=0)

( )( )

( )( )

( )( )

( )( )

2 2

2 2

*

2 2

2 2

1 1 1 2 1 1

1

1

1

1

1

2

R I

R I R I R R I

R I R I R R

R I

R I

I

R I

The is given by

R r r

n i n n i n n n nn i n n i n n n n

power refl

n i nr

n i

ectance R

n

n nR

n n

∴

=

⎡ ⎤ ⎡ ⎤− − − + ⎛ ⎞− + += =⎢ ⎥ ⎢ ⎥ ⎜ ⎟+ − + + + + +⎝ ⎠

− +=

+

− −=

+ −

⎦

+

⎣ ⎣ ⎦

2 2

2 2

2 2 2

2 2 2

cos sin 11cos sin

cos sin 11cos sin

i iTE

i i

i iTM

i i

nrn

n n

n

n

nr

nn n

θ θ

θ θ

θ θ

θ θ

− − −= =

++ −

− + − −= =

++ −

%

%

% % %

%

%

% %

%

, 0 :iAt normal incidence θ = ° At normal incidence(from Hecht, page 113)

λ

visible